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Structure of a Morse form foliation on a closed surface in terms of genus. (English) Zbl 1223.57022
Let $$\omega$$ be a closed $$1$$-form with Morse singularities on a genus $$g$$ closed surface $$M$$. The form $$\omega$$ defines a foliation $$\mathcal{F} _{\omega}$$ on $$M\backslash Sing\;\omega$$ and a singular foliation $$\overline{\mathcal{F}}_{\omega}$$ on the whole $$M$$. A leaf $$\gamma \in\mathcal{F}_{\omega}$$ is compactifiable if $$\gamma\cup Sing\;\omega$$ is compact. Non-compactifiable leaves of $$\mathcal{F}_{\omega}$$ form minimal components $$\mathcal{C}_{j}^{\min};$$ each such leaf is dense in its minimal component. If $$\gamma$$ is a compact singular leaf, a small closed tubular neighborhood of $$\gamma$$ is denoted by $$V(\gamma)$$.
The main result of this work is the relation $$c(\omega)+\sum_{\gamma}g(V(\gamma))+g(\bigcup \overline{\mathcal{C}_{j}^{\min}})=g,$$ where $$c(\omega)$$ is the number of homologically independent compact leaves. This result allows the author to prove a criterion for compactness of $$\overline{\mathcal{F}}_{\omega}$$, to estimate the number of its minimal components and to give an upper bound on the rank $$rk\;\omega,$$ in terms of genus.

MSC:
 57R30 Foliations in differential topology; geometric theory 58K65 Topological invariants on manifolds
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